Exchange interactions in Fe/Y multilayers

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Exchange interactions in Fe/Y multilayers

R. Elkabil

^a^,∗

, I. Elkaidi

^b^,^c

, F. Annouar

^b

, H. Lassri

^b

, A. Hamdoun

^a

, B. Bensassi

^c

, A. Berrada

^d

, R. Krishnan

^e

aLaboratoire de traitement d’information, Facult´e des Sciences Ben’Msik Sidi-Othmane, Casablanca, Morocco

bLaboratoire de Physique des Matériauxet de Micro-électronique, Faculté des Sciences Ain Chock, Université Hassan II, B.P. 5366 Mâarif, Route d’El Jadida, Km-8, Casablanca, Morocco

cUFR, Automatique et Informatique Industriel, Faculté des Sciences Ain Chock, Université Hassan II, B.P. 5366 Mâarif, Route d’El Jadida, Km-8, Casablanca, Morocco

dLaboratoire de Physique des Matériaux, Faculté des Sciences Université Mohammed V, B.P. 1014 Rabat, Morocco

eLaboratoire de Magn´etisme et d’Optique, URA 1531, 45 Avenue des Etats Unis, 78035 Versailles Cedex, France

Received 13 October 2004; received in revised form 1 November 2004; accepted 1 November 2004 Available online 25 December 2004

Abstract

The magnetization of Fe/Y multilayers has been measured as a function of temperature. A bulk-like T^3/2temperature dependence of the magnetization is observed for all multilayers in the temperature range 5–300 K. The spin-wave constant B is found to decrease inversely with tFe. A simple theoretical model with exchange interactions only, and with non-interacting magnons, has been used to explain the temperature dependence of the magnetization and the approximate values for the bulk exchange interaction Jb, surface exchange interaction Jsand the interlayer exchange interaction JIfor various Fe layer thicknesses have been obtained.

Keywords: Fe/Y multilayers; Magnetization; Spin-wave excitations; Exchange interactions

1. Introduction

Thin films and quasi-two-dimensional materials have many technological applications, including uses in electron- ics, data storage, and catalysis in the case of metal-on-metal films. Recent advance in film growth techniques such as molecular beam epitaxy (MBE) and in characterization meth- ods such as the surface magneto-optical Kerr effect provide not only an opportunity for further technological application but also allow one to consider thin films as an important testing ground of our understanding of atomic interactions.

The development of materials with characteristics tailored to a specific application requires a detailed understanding of their microscopic interactions, how these interactions are affected by factors such as composition and preparation, and how they determine the material properties. For example, the

∗Corresponding author.

E-mail address: relkabil@yahoo.fr (R. Elkabil).

use of ultrathin magnetic films for data storage requires that the magnetization of the film be set and read with a high degree of accuracy and spatial resolution.

One common approach to investigate the magnetic behavior of two-dimensional systems is to study the spontaneous magnetization, its temperature dependence and the Curie temperature, T_C, of ferromagnetic thin films. Exper- imentally, a drastic change of the magnetic behavior is found for films thinner than a critical thickness which varies from author to author[1–5]. The basic problem is whether these observations describe intrinsic properties of two-dimensional ferromagnets or are mainly produced by structural imperfections (contamination, diffusion, island growth).

Recently, Pinettes and Lacroix examine the influence of the anisotropy on the thickness dependence of the spin-wave excitation spectra and calculate the thermal variation of the magnetization as a function of film thickness[6]. It has been shown that the surface anisotropy affect the thickness dependence of the magnetization. In the case of a multilayer the

doi:10.1016/j.jallcom.2004.11.010

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thermal variation of the magnetization and the Curie temperature is also affected by the interlayer exchange coupling [7,8]. Furthermore, interdiffusion causing graded interfaces and disorder resulting in a distribution of exchange interactions, can play an important role. Especially when the magnetic layer thickness is in the monolayer regime the latter two factors might dominate the temperature dependence.

In this paper we present our studies on Fe/Y multilayers prepared by the evaporation method. Assuming a Heisenberg framework, we propose a simple theoretical model, including both intralayer and interlayer exchange interactions. The exchange interactions strengths are adjusted directly by fitting the calculated magnetization to the measured one.

2. Experimental

The samples were evaporated by using a dual electron- beam source. The pressure was maintained in the range of (3–5)×10⁻⁹Torr. The deposition rate of 0.3–0.5 ˚A/s was applied and each layer thickness was determined by quartz monitors. The glass substrates were mounted on the copper block cooled below 200 K in order to reduce atomic diffusion at interfaces. The Y layer thickness was held constant at about 20 ˚A for all the samples, and the Fe layer thickness was varied between 20 and 50 ˚A. The magnetization M was measured using a vibrating sample magnetometer (VSM).

3. Results and discussion

Fig. 1shows the temperature dependence of M for several values of t_Fethicknesses. It can be noticed that T_Cdecreases when tFedecreases. The low-temperature magnetization was

Fig. 1. The temperature dependence of the magnetization for Fe thicknesses.

All data were fitted using Eq.(10)as indicated by solid lines.

studied in detail for a few samples. For three-dimensional magnetic films, the magnetization has a T^3/2dependence due to classical spin-wave excitations. In such cases, according to spin-wave theory, the temperature dependence should follow the relation:

M(5 K)−M(T)

M(5 K) =BT³^/² (1)

In all cases this behavior is observed for temperatures as high as T_C/3. The spin-wave constant B decreases from 16×10⁻⁶K⁻^3/2 for t_Fe= 20 ˚A to 9×10⁻⁶K⁻^3/2 for tFe= 50 ˚A. These values are much larger than the value of 5×10⁻⁶K⁻^3/2found for bulk Fe. The B versus 1/tFeis plot- ted for the samples with 20≤tFe≤50 ˚A inFig. 2. It is seen that the experimental points align well in a straight line. The values extrapolated to 1/tFe= 0 are in good agreement with those found for the bulk Fe. It was observed that the param- eters B in Eq.(1)depend on t_Feaccording to:

B(tFe)=B_∞+Bs

tFe

(2) where B_∞is the bulk spin-wave parameter of Fe and B_ssur- face B value. The linear relation between the spin-wave pa- rameter B and the reciprocal of the magnetic film thickness was reported also by Gradmann and co-workers[9,10]for Fe(1 1 0) films on W(1 1 0).

The apparent universality of Bloch’s T^3/2-law for the temperature dependence of the spontaneous magnetization, and of generalization thereof, is considered by many others [11–13]. It is argued that in the derivation one should not only consider the exchange interaction between the spin, but also the other interactions between them, leading to elliptical spin precession and deviations from the parabolic dispersion of magnons. Also interaction effects are important to explain

Fig. 2. ThetFe⁻¹dependence of the B.

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Fig. 3. The tFedependence of M×tFefor T = 5 and 300 K.

the apparent universality of generalized Bloch law exponents n, defined by M(T) = M(0)−const.Tⁿ, valid in wide temper- ature range, and for dimensionality d = 1, 2, and 3. However, due to interactions, the Bloch exponent n depends not only on the dimensionality d of the system, but also on the spin quantum number S of the system[13].

Besides it is known from the dead layer phenomenologi- cal mode1 that the magnetization of a multilayer for a given temperature can be expressed as

M=M0

1−2δ

t

(3) where M0is the bulk magnetization value, andδis the dead layer thickness at each side of the magnetic layer. So if we plot M×t as a function of the magnetic layer thickness t, we obtain a straight line, the slope of which gives M₀and the intercept 2δ.Fig. 3shows the results at 5 and 300 K. It can be seen that the two slopes are practically the same and correspond to M₀= (1660±40) emu/cm³, which agrees well with the bulk value for Fe. The intercept of the straight lines give the dead layer thickness 2δof about 10 ˚A.

To describe the experiment results, we attempted to use a simple model. We suppose that the multilayer (X_n/Y_m)_qis formed by an alternate deposition of a magnetic layer (X) and non-magnetic one (Y). The multilayer is characterized by the number (q) of pairs (X/Y), the number (n) of atomic planes in the magnetic layerµand the number (m) of atomic planes in the non-magnetic layer. We chose the lattice unit vectors (e_x, e_y, e_z) so that e_z is perpendicular to the atomic planes. We note by S_i_αµthe vector spin operator of the atom i (i = 1, 2,. . ., N) in the planeα(α= 1, 2,. . ., n) of the mag- netic layer µ(µ= 1, 2,. . ., q). Further we suppose that the multilayer is characterized by a rigid lattice and by perfectly sharp layer interfaces without structural imperfections (con-

tamination, diffusion, island growth, etc.). In this framework the Heisenberg-type system Hamiltonian is given by:

H = −Jb



 ^b

iαµ,jαµ

S_iαµS_jαµ+

iαµ,jαµ

S_iαµS_jα_µ





−Js

s iαµ,jαµ

SiαµSjαµ−JI

I iαµ,jαµ

SiαµS_jα_µ (4)

H describes the exchange interactions in the same magnetic layer (bulk and surface) as well as the exchange interactions between adjacent magnetic layers. J_band J_sare the bulk and surface exchange interactions, J_Iis the strength of the interlayer coupling restricted to the surface layers. This model is equivalent to a three-dimensional anisotropic ferromagnetic.

Further we denote by_Ξ

the summation on the sites of the bulk layer planes (Ξ= b), surface layer planes (Ξ= s) or the surfaces planes coupled via the non-magnetic layer (Ξ= I).

The symbol ·denotes the pairs of nearest-neighbors atoms or adjacent magnetic planes.

In the Holstein–Primakoff formulation[14], the creation and annihilation operators (a⁺_iαµ andaiαµ) for each atomic spin are related to the spin operators by:

S_iαµ^X +iS_iαµ^Y =(2S)¹^/²f_iαµ(2S)a_iαµ and S_iαµ^X −iS^Y_iαµ

=(2S)¹^/²a⁺_iαµf_iαµ(2S) (5) In the framework of non-interacting spin-wave theory, the linear approximation of the Holstein–Primakoff method is sufficient to describe the main magnetic behavior and the cor- rection terms are quite-small at low temperatures (T < TC/3) [15,16]. So, the value of fiαµ(2S) is fixed to 1.

We pass from the atomic variables to the magnon variables after a two-dimensional Fourier transformation:

H =H0+ s k,αµ

A_kb⁺_kαµb_kαµ+ b k,αµ

B_kb⁺_kαµb_kαµ

+

k, αµ,αµ

C_kb⁺_kαµb_kα_µ+ I k, αµ,αµ

D_kb⁺_kαµb_kα_µ (6) where

A_k =(Js(2n−(λ⁺_k +λ⁻_k))+2Jbn^⊥)S+2JInS, Bk =(4n^⊥+2n−(λ⁺_k +λ⁻_k))JbS,

Ck = −JbSλ_k, Dk= −JISλ_k (7) H0is a constant term, the coefficientsλ⁺_k,λ⁻_k,λ_kandλ_kde- pend on the crystallographic structure of the magnetic layer.

n represent the number of nearest-neighbors sites in the same atomic plane, while n^⊥(n) is the number of nearest- neighbors in the adjacent plane in the same (adjacent) mag- netic layer. For bcc(1 1 0) (n= 4 and n^⊥= 2) with the lattice constant a and in the case where the non-magnetic layer do

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not disturb the succession order of the magnetic atomic planes (n=2):

λ⁺_k =λ⁻_k =4 cos akx

√2 2

cos

ak_y 2

, λ_k=λ_k =4 cos

aky

2

(8) The spin system is characterized by 2nq×2nq equations, then the resulting secular equation is given by a (2nq×2nq) matrix:

W²^nq×²^nq= U⁽^nq×nq⁾

−U⁽^nq×nq⁾

where

U⁽^nq×nq⁾=







U₁⁽^n×n⁾ U₂⁽^n×n⁾

U₃⁽^n×n⁾ U₁⁽^n×n⁾ U₂⁽^n×n⁾

· · ·

U₃⁽^n×n⁾ U₁⁽^n×n⁾ U₂⁽^n×n⁾ U₃⁽^n×n⁾ U₁⁽^n×n⁾





 ,

U₁⁽^n×n⁾=







Bk Dk

D_k C_k D_k

. · ·

· · ·

· · · D_k C_k D_k

D_k B_k







, U₂⁽^n×n⁾=







0 . . . 0

. . .

0 . .

E_k0 . . . 0







and U₃⁽^n×n⁾=







0 . . . 0E_k

. . 0

. . .

0 . . . 0





 (9)

Among the 2(n×q) eigenvalues of the matrix W(2nq×2nq), we consider the n×q positive ones which correspond to the n×q magnon excitation branchesω^r_k(r = 1, 2,. . ., n×q).

Theses branches can be classified into n groups of q quasi- degenerate components in the usual case where JIremain suf- ficiently small compared to the effective intralayer exchange strength (Fig. 4). The reduced magnetization versus temperature is computed numerically from:

m(T)=1− 1 N_knqS

k,r

1

exp(ω_k^r/kBT)−1 (10) The coefficient N_kindicates the number of k points taken in the first Brouillin zone.

Using Eq.(10), satisfactory fits were obtained for the m(T) data for all of the Fe/Y multilayer films. The m(T) theory curves obtained from the fits are shown inFig. 1, well match- ing the experimental data points. The values of J_b, J_s, J_Iob- tained from the fits are listed inTable 1for all films (taken S = 1). The derived bulk exchange interaction constants all consistently fall in the range expected for the exchange interaction of bulk Fe[17,18]. In the literature, the order of the

Table 1

The fitting results from Eq.(10)for Fe(tFe)/Y(tY= 20 ˚A)

tFe( ˚A) Jb/kB(K) Js/kB(K) JI/kB(K)

20 141 69 1.1

25 142 67 1

40 140 66 0.9

50 141 66 1

Jbis the bulk exchange interaction between neighboring Fe atoms, Jsis the surface exchange interaction; JIis the interlayer coupling.

interlayer coupling strengths are diverse[19–21]. For this parameter, our results remain in the same order but lesser than that reported by Gutierrez et al. in Fe/Ag multilayers[22].

Fig. 4. Spin-wave excitation spectrum vs. kx(ky=kx√

2) for bcc(1 1 0) ferro- magnetic multilayer with q = 20; n = 3; Js= 40 K; Jb= 100 K; JI= 5 K; S = 1.

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4. Conclusions

In conclusion, the temperature dependence of the magnetization of Fe/Y multilayers has been investigated for various Fe layer thicknesses. The thermal variation of the magnetization in ferromagnetic multilayer films is calculated within the framework of non-interacting spin-wave theory. The pro- posed model neglects all kind of interactions except the exchange interactions, even though it is in a satisfactory agreement with the experimental data. This simple model has al- lowed us to obtain numerical estimates for the bulk exchange interaction J_b, surface exchange interaction J_sand the inter- layer coupling strength J_Iat various Fe/Y multilayers. Com- pared to the bulk exchange interaction, however, the interlayer coupling is considerably weak.

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